Namespaces
Variants
Actions

Difference between revisions of "Galois theory"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
Line 1: Line 1:
In the most general sense it is a theory dealing with mathematical objects on the basis of their automorphism groups. For instance, Galois theories of fields, rings, topological spaces, etc., are possible. In a narrower sense Galois theory is the Galois theory of fields. The theory originated in the context of finding roots of algebraic equations of high degrees. The familiar formula for solving equations of degree two dates back to early Antiquity. Methods for solving cubic (cf. [[Cardano formula|Cardano formula]]) and quartic (cf. [[Ferrari method|Ferrari method]]) equations were discovered in the 16th century. Unsuccessful attempts to find formulas for solving quintic and higher-degree equations were made during the three centuries which followed. It was finally proved by N.H. Abel in 1824 that there are no solutions in radicals of the general equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431601.png" />. The next problem was to find necessary and sufficient conditions to be satisfied by the coefficients of an equation for the latter to be solvable in radicals, i.e. for it to be reducible to a chain of two-term equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431602.png" />. This problem was solved by E. Galois; his results were exposed in a letter on the eve of his death (1832), and published in 1846. The theory of Galois will now be summarized in modern language.
+
In the most general sense it is a theory dealing with mathematical
 +
objects on the basis of their automorphism groups. For instance,
 +
Galois theories of fields, rings, topological spaces, etc., are
 +
possible. In a narrower sense Galois theory is the Galois theory of
 +
fields. The theory originated in the context of finding roots of
 +
algebraic equations of high degrees. The familiar formula for solving
 +
equations of degree two dates back to early Antiquity. Methods for
 +
solving cubic (cf.
 +
[[Cardano formula|Cardano formula]]) and quartic (cf.
 +
[[Ferrari method|Ferrari method]]) equations were discovered in the
 +
16th century. Unsuccessful attempts to find formulas for solving
 +
quintic and higher-degree equations were made during the three
 +
centuries which followed. It was finally proved by N.H. Abel in 1824
 +
that there are no solutions in radicals of the general equation of
 +
degree $\ge 5$. The next problem was to find necessary and sufficient
 +
conditions to be satisfied by the coefficients of an equation for the
 +
latter to be solvable in radicals, i.e. for it to be reducible to a
 +
chain of two-term equations of the form $x^n-a=0$. This problem was solved
 +
by E. Galois; his results were exposed in a letter on the eve of his
 +
death (1832), and published in 1846. The theory of Galois will now be
 +
summarized in modern language.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431603.png" /> be an arbitrary field. An extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431604.png" /> is any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431605.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431606.png" /> as a subfield. Any extension may be regarded as a linear space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431607.png" />; if this space has finite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431608.png" />, the extension is called finite, while the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g0431609.png" /> is called the degree of the extension. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316010.png" /> of an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316011.png" /> is said to be algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316012.png" /> if it is the root of an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316014.png" /> is a non-zero polynomial with coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316015.png" /> (this polynomial may be taken to be irreducible). The smallest extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316016.png" /> containing the algebraic element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316017.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316018.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316019.png" />. A finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316021.png" /> is called separable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316022.png" /> and if the irreducible polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316024.png" /> as a root has no multiple roots. If the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316025.png" /> is zero (e.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316026.png" /> is a number field), any finite extension is separable (the theorem on the primitive element). The splitting field of an irreducible polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316028.png" /> is the smallest extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316029.png" /> that contains all the roots of this polynomial. The degree of such an extension is divisible by the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316030.png" /> and is equal to it if all the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316031.png" /> can be expressed as polynomials in one of them. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316033.png" /> is called normal if it is the splitting field of a certain polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316034.png" />, and is called a Galois extension if it is normal and separable. The group of all automorphisms of a Galois extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316035.png" /> that leave all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316036.png" /> invariant is called the Galois group of this extension and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316037.png" />. Its order (the number of elements) is equal to the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316038.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316039.png" />. To each subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316041.png" /> corresponds a subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316043.png" />, consisting of all elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316044.png" /> that remain fixed under all automorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316045.png" />. Conversely, to each subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316046.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316047.png" /> corresponds a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316049.png" />. It consists of all automorphisms leaving each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316050.png" /> invariant. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316051.png" /> is a Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316053.png" />. The main theorem in Galois theory states that these correspondences are mutually inverse, and are therefore one-to-one correspondences between all subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316054.png" /> and all subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316055.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316056.png" />. Thus, the description of all subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316057.png" /> is reduced to the description of all subgroups of the finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316058.png" />, which is a much simpler task. It is important to note that in this correspondence certain "good" properties of subgroups correspond to the "good" properties of subfields and vice versa. Thus, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316059.png" /> will be a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316060.png" /> if and only if the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316061.png" /> which corresponds to it is a Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316062.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316063.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316064.png" />. To each ascending chain
+
Let $k$ be an arbitrary field. An extension of $k$ is any field $K$
 +
that contains $k$ as a subfield. Any extension may be regarded as a
 +
linear space over $k$; if this space has finite dimension $n$, the
 +
extension is called finite, while the dimension $n$ is called the
 +
degree of the extension. An element $\def\a{\alpha}\a$ of an extension of $k$ is said
 +
to be algebraic over $k$ if it is the root of an equation $f=0$, where
 +
$f$ is a non-zero polynomial with coefficients from $k$ (this
 +
polynomial may be taken to be irreducible). The smallest extension of
 +
$k$ containing the algebraic element $\a$ over $k$ is usually denoted
 +
by $k(\a)$. A finite extension $K$ of $k$ is called separable if $K=k(\a)$ and
 +
if the irreducible polynomial $f$ with $\a$ as a root has no multiple
 +
roots. If the characteristic of $k$ is zero (e.g. if $k$ is a number
 +
field), any finite extension is separable (the theorem on the
 +
primitive element). The splitting field of an irreducible polynomial
 +
$f$ in $k[X]$ is the smallest extension of $k$ that contains all the
 +
roots of this polynomial. The degree of such an extension is divisible
 +
by the degree of $f$ and is equal to it if all the roots of $f$ can be
 +
expressed as polynomials in one of them. An extension $K$ of $k$ is
 +
called normal if it is the splitting field of a certain polynomial in
 +
$k[X[$, and is called a Galois extension if it is normal and
 +
separable. The group of all automorphisms of a Galois extension $K$
 +
that leave all elements of $k$ invariant is called the Galois group of
 +
this extension and is denoted by $\def\Gal{\textrm{Gal}}\Gal(K/k)$. Its order (the number of
 +
elements) is equal to the degree of $K$ over $k$. To each subgroup $H$
 +
of $\Gal(K/k)$ corresponds a subfield $P$ of $K$, consisting of all elements
 +
from $K$ that remain fixed under all automorphisms from
 +
$H$. Conversely, to each subfield $P\subset K$ that contains $k$ corresponds a
 +
subgroup $H$ of $\Gal(K/k)$. It consists of all automorphisms leaving each
 +
element of $P$ invariant. Here, $K$ is a Galois extension of $P$ and
 +
$\Gal(K/P)=H$. The main theorem in Galois theory states that these
 +
correspondences are mutually inverse, and are therefore one-to-one
 +
correspondences between all subgroups of $\Gal(K/k)$ and all subfields of $K$
 +
containing $k$. Thus, the description of all subfields of $K$ is
 +
reduced to the description of all subgroups of the finite group $\Gal(K/k)$,
 +
which is a much simpler task. It is important to note that in this
 +
correspondence certain "good" properties of subgroups correspond to
 +
the "good" properties of subfields and vice versa. Thus, a subgroup
 +
$H$ will be a normal subgroup of $\Gal(K/k)=G$ if and only if the field $P$
 +
which corresponds to it is a Galois extension of $k$. Moreover, $\Gal(P/k)$ is
 +
isomorphic to $G/H$. To each ascending chain
 +
$$k=K_0,\subset K_1\subset\cdots\subset K_r=K\label{1}$$
 +
of subfields of $K$
 +
corresponds a descending chain
 +
$$G=H_0,\supset H_1\supset\cdots\supset H_r=\{e\}\label{2}$$
 +
of subgroups of $G$, where
 +
$H_i=\Gal(K/K_i)$. The chain (2) is a normal series (i.e. each group $H_{i+1}$ is a normal
 +
subgroup of $H_i$ if $0\le i < r$) if and only if each field $K_{i+1}$ in the chain (1)
 +
is a Galois extension of $K_i$, and in this a case one has $H_i/H_{i+1} \cong \Gal(K_{i+1}/K_i)$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
These results are applied to the solution of algebraic equations as
 +
follows. Let $f$ be an irreducible polynomial without multiple roots
 +
over the field $k$ and let $K$ be its splitting field (which will be a
 +
Galois extension of $k$). The Galois group of this extension is called
 +
the Galois group of the equation $f=0$. Solving the equation $f=0$ is
 +
reduced to solving a chain of equations $f_1=0,\dots,f_r=0$ if and only if $K$ is
 +
contained in a field $\bar K$ that is the last term of the ascending chain
 +
of fields
 +
$$k=K_0,\subseteq K_1\subseteq\cdots\subseteq K_r=K,$$
 +
where $K_i$, $i=1,\dots,r$, is the splitting field of $f_i$ over
 +
$K_{i-1}$. This last condition is equivalent with the group $G=\Gal(K/k)$ being a
 +
quotient group of the group $\bar G=\Gal(\bar K/k)$ with a normal series whose factors $H_i/H_{i+1}$
 +
are isomorphic to the Galois groups of the equations $f_i=0$ over $K_{i-1}$.
  
of subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316066.png" /> corresponds a descending chain
+
Let the field $k$ contain all roots of unity of order $n$. Then, for
 +
any $a\in k$, the splitting field of the polynomial $x^n-a$ is $k(\a)$, where $\a$
 +
is one of the values of the radical $a^{1/n}$. In such a case $\Gal(k(\a)/k)$ is a
 +
cyclic group of order dividing $n$; conversely, if $\Gal(K/k)$ is a cyclic
 +
group of order $n$, one has $K=k(\a)$, where $\a$ is the root of some
 +
two-term equation $x^n-a = 0$. Thus, if $k$ contains the roots of unity of all
 +
possible orders, then the equation $f=0$ is solvable in radicals if and
 +
only if its Galois group is solvable (i.e. has a normal series with
 +
cyclic factors $H_i/H_{i+1}$). This condition of solvability in radicals is also
 +
valid if $k$ does not contain all possible roots of unity, since the
 +
Galois group $\Gal(k'/k)$ of the extension $k'$ which is obtained by adjoining
 +
these roots is always solvable.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
In practical applications of the solvability condition, a very
 +
important fact is that the Galois group of an equation can be computed
 +
without solving the equation itself. The idea of this computation can
 +
be stated as follows. Each automorphism of the splitting field of a
 +
polynomial $f$ induces a permutation of its roots, and the
 +
automorphism is completely determined by this permutation. For this
 +
reason the Galois group of the equation may be treated, in principle,
 +
as a subgroup of the group of permutations of its roots (namely, the
 +
subgroup consisting of permutations that preserve all algebraic
 +
relations between the roots). The relations between the roots of a
 +
polynomial yield certain relations between its coefficients (by virtue
 +
of Viète's formulas); by analyzing these relations it is possible to
 +
determine the relations between the roots of the polynomial and thus
 +
to compute the Galois group of the equation. In the general case the
 +
Galois group of an algebraic equation consists of all permutations of
 +
the roots, i.e. is the symmetric group of degree $n$. Since a
 +
symmetric group is unsolvable for $n\ge 5$, there are, generally speaking,
 +
no solutions in radicals of quintic equations and equations of a
 +
higher degree (Abel's theorem).
  
of subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316069.png" />. The chain (2) is a normal series (i.e. each group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316070.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316071.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316072.png" />) if and only if each field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316073.png" /> in the chain (1) is a Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316074.png" />, and in this a case one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316075.png" />.
+
The ideas of Galois theory permit, in particular, to give a complete
 +
description of the class of construction problems that are solvable by
 +
ruler and compass. It is possible to show by methods of analytic
 +
geometry that any such construction problem can be reduced to some
 +
algebraic equation over the field of rational numbers, and the problem
 +
is solvable by using a ruler and compass if and only if the
 +
corresponding equation is solvable in quadratic radicals. This is the
 +
case if and only if the Galois group of the equation has a normal
 +
series whose factors are groups of order two, and this occurs if and
 +
only if its order is a power of two. Thus, a construction problem
 +
solvable by ruler and compass is reduced to solving an equation whose
 +
splitting field has degree $2^s$ over the field of rational numbers for
 +
some $s$; if the degree of the equation is not of the form $2^s$, such a
 +
construction is impossible. This is the case for the problem of
 +
doubling the cube (which leads to the cubic equation $x^3-2 = 0$) and for the
 +
problem of trisecting a given angle (which also leads to a cubic
 +
equation). If $p$ is a prime number, the problem of constructing a
 +
regular $p$-gon leads to the equation $x_{p-1}+\cdots + x + 1 = 0$, whose splitting field is
 +
generated by any root and is therefore of degree $p-1$, the degree of
 +
the equation. In this case a ruler-and-compass construction is
 +
possible if and only if $p=2^s+1$ (e.g. if $p=5$ and $p=17$ it is possible, but
 +
is not possible for $p=7$ or $p=13$).
  
These results are applied to the solution of algebraic equations as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316076.png" /> be an irreducible polynomial without multiple roots over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316077.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316078.png" /> be its splitting field (which will be a Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316079.png" />). The Galois group of this extension is called the Galois group of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316080.png" />. Solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316081.png" /> is reduced to solving a chain of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316082.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316083.png" /> is contained in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316084.png" /> that is the last term of the ascending chain of fields
+
The ideas of Galois had a decisive influence on the development of
 
+
algebra during almost a whole century. Galois theory was extended and
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316085.png" /></td> </tr></table>
+
generalized in many directions. W. Krull developed a Galois theory for
 
+
infinite extensions (cf.
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316087.png" />, is the splitting field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316088.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316089.png" />. This last condition is equivalent with the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316090.png" /> being a quotient group of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316091.png" /> with a normal series whose factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316092.png" /> are isomorphic to the Galois groups of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316093.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316094.png" />.
+
[[Galois topological group|Galois topological group]]); it was proved
 
+
(the Kronecker–Weber theorem) that the roots of an equation with
Let the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316095.png" /> contain all roots of unity of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316096.png" />. Then, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316097.png" />, the splitting field of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316098.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g04316099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160100.png" /> is one of the values of the radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160101.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160102.png" /> is a cyclic group of order dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160103.png" />; conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160104.png" /> is a cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160105.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160107.png" /> is the root of some two-term equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160108.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160109.png" /> contains the roots of unity of all possible orders, then the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160110.png" /> is solvable in radicals if and only if its Galois group is solvable (i.e. has a normal series with cyclic factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160111.png" />). This condition of solvability in radicals is also valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160112.png" /> does not contain all possible roots of unity, since the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160113.png" /> of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160114.png" /> which is obtained by adjoining these roots is always solvable.
+
rational coefficients and with an Abelian Galois group are rational
 
+
linear combinations of roots of unity; a classification of the Abelian
In practical applications of the solvability condition, a very important fact is that the Galois group of an equation can be computed without solving the equation itself. The idea of this computation can be stated as follows. Each automorphism of the splitting field of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160115.png" /> induces a permutation of its roots, and the automorphism is completely determined by this permutation. For this reason the Galois group of the equation may be treated, in principle, as a subgroup of the group of permutations of its roots (namely, the subgroup consisting of permutations that preserve all algebraic relations between the roots). The relations between the roots of a polynomial yield certain relations between its coefficients (by virtue of Viète's formulas); by analyzing these relations it is possible to determine the relations between the roots of the polynomial and thus to compute the Galois group of the equation. In the general case the Galois group of an algebraic equation consists of all permutations of the roots, i.e. is the symmetric group of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160116.png" />. Since a symmetric group is unsolvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160117.png" />, there are, generally speaking, no solutions in radicals of quintic equations and equations of a higher degree (Abel's theorem).
+
extensions of a given algebraic number field (class field theory) was
 
+
given; a proof was given for the existence of an algebraic number
The ideas of Galois theory permit, in particular, to give a complete description of the class of construction problems that are solvable by ruler and compass. It is possible to show by methods of analytic geometry that any such construction problem can be reduced to some algebraic equation over the field of rational numbers, and the problem is solvable by using a ruler and compass if and only if the corresponding equation is solvable in quadratic radicals. This is the case if and only if the Galois group of the equation has a normal series whose factors are groups of order two, and this occurs if and only if its order is a power of two. Thus, a construction problem solvable by ruler and compass is reduced to solving an equation whose splitting field has degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160118.png" /> over the field of rational numbers for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160119.png" />; if the degree of the equation is not of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160120.png" />, such a construction is impossible. This is the case for the problem of doubling the cube (which leads to the cubic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160121.png" />) and for the problem of trisecting a given angle (which also leads to a cubic equation). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160122.png" /> is a prime number, the problem of constructing a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160124.png" />-gon leads to the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160125.png" />, whose splitting field is generated by any root and is therefore of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160126.png" />, the degree of the equation. In this case a ruler-and-compass construction is possible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160127.png" /> (e.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160128.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160129.png" /> it is possible, but is not possible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160130.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160131.png" />).
+
field with given solvable Galois group over the field of rational
 
+
numbers (cf.
The ideas of Galois had a decisive influence on the development of algebra during almost a whole century. Galois theory was extended and generalized in many directions. W. Krull developed a Galois theory for infinite extensions (cf. [[Galois topological group|Galois topological group]]); it was proved (the Kronecker–Weber theorem) that the roots of an equation with rational coefficients and with an Abelian Galois group are rational linear combinations of roots of unity; a classification of the Abelian extensions of a given algebraic number field (class field theory) was given; a proof was given for the existence of an algebraic number field with given solvable Galois group over the field of rational numbers (cf. [[Galois theory, inverse problem of|Galois theory, inverse problem of]]). Nevertheless, classical Galois theory still contains many unsolved problems. For instance, it is not known whether any finite group occurs as the Galois group of an equation over the field of rational numbers.
+
[[Galois theory, inverse problem of|Galois theory, inverse problem
 +
of]]). Nevertheless, classical Galois theory still contains many
 +
unsolved problems. For instance, it is not known whether any finite
 +
group occurs as the Galois group of an equation over the field of
 +
rational numbers.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Galois,   "Écrits et mémoires d'E. Galois" , Gauthier-Villars (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.G. [N.G. Chebotarev] Tschebotaröw,   "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N.G. Chebotarev,   "Galois theory" , Moscow-Leningrad (1936) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.M. Postnikov,   "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.M. Postnikov,   "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S. Lang,   "Algebra" , Addison-Wesley (1974)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Koch,   "Galoissche Theorie der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043160/g043160132.png" />-Erweiterungen" , Deutsch. Verlag Wissenschaft.  (1970)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E. Artin,   "Galois theory" , Notre Dame Univ. , Indiana (1948)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD>
 +
<TD valign="top"> E. Galois, "Écrits et mémoires d'E. Galois" , Gauthier-Villars (1962)</TD>
 +
</TR><TR><TD valign="top">[2]</TD>
 +
<TD valign="top"> N.G. [N.G. Chebotarev] Tschebotaröw, "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[3]</TD>
 +
<TD valign="top"> N.G. Chebotarev, "Galois theory" , Moscow-Leningrad (1936) (In Russian)</TD>
 +
</TR><TR><TD valign="top">[4]</TD>
 +
<TD valign="top"> M.M. Postnikov, "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[5]</TD>
 +
<TD valign="top"> M.M. Postnikov, "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[6]</TD>
 +
<TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD>
 +
</TR><TR><TD valign="top">[7]</TD>
 +
<TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974)</TD>
 +
</TR><TR><TD valign="top">[8]</TD>
 +
<TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)</TD>
 +
</TR><TR><TD valign="top">[9]</TD>
 +
<TD valign="top"> H. Koch, "Galoissche Theorie der $p$-Erweiterungen" , Deutsch. Verlag Wissenschaft.  (1970)</TD>
 +
</TR><TR><TD valign="top">[10]</TD>
 +
<TD valign="top"> E. Artin, "Galois theory" , Notre Dame Univ. , Indiana (1948)</TD>
 +
</TR></table>

Revision as of 17:39, 21 November 2011

In the most general sense it is a theory dealing with mathematical objects on the basis of their automorphism groups. For instance, Galois theories of fields, rings, topological spaces, etc., are possible. In a narrower sense Galois theory is the Galois theory of fields. The theory originated in the context of finding roots of algebraic equations of high degrees. The familiar formula for solving equations of degree two dates back to early Antiquity. Methods for solving cubic (cf. Cardano formula) and quartic (cf. Ferrari method) equations were discovered in the 16th century. Unsuccessful attempts to find formulas for solving quintic and higher-degree equations were made during the three centuries which followed. It was finally proved by N.H. Abel in 1824 that there are no solutions in radicals of the general equation of degree $\ge 5$. The next problem was to find necessary and sufficient conditions to be satisfied by the coefficients of an equation for the latter to be solvable in radicals, i.e. for it to be reducible to a chain of two-term equations of the form $x^n-a=0$. This problem was solved by E. Galois; his results were exposed in a letter on the eve of his death (1832), and published in 1846. The theory of Galois will now be summarized in modern language.

Let $k$ be an arbitrary field. An extension of $k$ is any field $K$ that contains $k$ as a subfield. Any extension may be regarded as a linear space over $k$; if this space has finite dimension $n$, the extension is called finite, while the dimension $n$ is called the degree of the extension. An element $\def\a{\alpha}\a$ of an extension of $k$ is said to be algebraic over $k$ if it is the root of an equation $f=0$, where $f$ is a non-zero polynomial with coefficients from $k$ (this polynomial may be taken to be irreducible). The smallest extension of $k$ containing the algebraic element $\a$ over $k$ is usually denoted by $k(\a)$. A finite extension $K$ of $k$ is called separable if $K=k(\a)$ and if the irreducible polynomial $f$ with $\a$ as a root has no multiple roots. If the characteristic of $k$ is zero (e.g. if $k$ is a number field), any finite extension is separable (the theorem on the primitive element). The splitting field of an irreducible polynomial $f$ in $k[X]$ is the smallest extension of $k$ that contains all the roots of this polynomial. The degree of such an extension is divisible by the degree of $f$ and is equal to it if all the roots of $f$ can be expressed as polynomials in one of them. An extension $K$ of $k$ is called normal if it is the splitting field of a certain polynomial in $k[X[$, and is called a Galois extension if it is normal and separable. The group of all automorphisms of a Galois extension $K$ that leave all elements of $k$ invariant is called the Galois group of this extension and is denoted by $\def\Gal{\textrm{Gal}}\Gal(K/k)$. Its order (the number of elements) is equal to the degree of $K$ over $k$. To each subgroup $H$ of $\Gal(K/k)$ corresponds a subfield $P$ of $K$, consisting of all elements from $K$ that remain fixed under all automorphisms from $H$. Conversely, to each subfield $P\subset K$ that contains $k$ corresponds a subgroup $H$ of $\Gal(K/k)$. It consists of all automorphisms leaving each element of $P$ invariant. Here, $K$ is a Galois extension of $P$ and $\Gal(K/P)=H$. The main theorem in Galois theory states that these correspondences are mutually inverse, and are therefore one-to-one correspondences between all subgroups of $\Gal(K/k)$ and all subfields of $K$ containing $k$. Thus, the description of all subfields of $K$ is reduced to the description of all subgroups of the finite group $\Gal(K/k)$, which is a much simpler task. It is important to note that in this correspondence certain "good" properties of subgroups correspond to the "good" properties of subfields and vice versa. Thus, a subgroup $H$ will be a normal subgroup of $\Gal(K/k)=G$ if and only if the field $P$ which corresponds to it is a Galois extension of $k$. Moreover, $\Gal(P/k)$ is isomorphic to $G/H$. To each ascending chain $$k=K_0,\subset K_1\subset\cdots\subset K_r=K\label{1}$$ of subfields of $K$ corresponds a descending chain $$G=H_0,\supset H_1\supset\cdots\supset H_r=\{e\}\label{2}$$ of subgroups of $G$, where $H_i=\Gal(K/K_i)$. The chain (2) is a normal series (i.e. each group $H_{i+1}$ is a normal subgroup of $H_i$ if $0\le i < r$) if and only if each field $K_{i+1}$ in the chain (1) is a Galois extension of $K_i$, and in this a case one has $H_i/H_{i+1} \cong \Gal(K_{i+1}/K_i)$.

These results are applied to the solution of algebraic equations as follows. Let $f$ be an irreducible polynomial without multiple roots over the field $k$ and let $K$ be its splitting field (which will be a Galois extension of $k$). The Galois group of this extension is called the Galois group of the equation $f=0$. Solving the equation $f=0$ is reduced to solving a chain of equations $f_1=0,\dots,f_r=0$ if and only if $K$ is contained in a field $\bar K$ that is the last term of the ascending chain of fields $$k=K_0,\subseteq K_1\subseteq\cdots\subseteq K_r=K,$$ where $K_i$, $i=1,\dots,r$, is the splitting field of $f_i$ over $K_{i-1}$. This last condition is equivalent with the group $G=\Gal(K/k)$ being a quotient group of the group $\bar G=\Gal(\bar K/k)$ with a normal series whose factors $H_i/H_{i+1}$ are isomorphic to the Galois groups of the equations $f_i=0$ over $K_{i-1}$.

Let the field $k$ contain all roots of unity of order $n$. Then, for any $a\in k$, the splitting field of the polynomial $x^n-a$ is $k(\a)$, where $\a$ is one of the values of the radical $a^{1/n}$. In such a case $\Gal(k(\a)/k)$ is a cyclic group of order dividing $n$; conversely, if $\Gal(K/k)$ is a cyclic group of order $n$, one has $K=k(\a)$, where $\a$ is the root of some two-term equation $x^n-a = 0$. Thus, if $k$ contains the roots of unity of all possible orders, then the equation $f=0$ is solvable in radicals if and only if its Galois group is solvable (i.e. has a normal series with cyclic factors $H_i/H_{i+1}$). This condition of solvability in radicals is also valid if $k$ does not contain all possible roots of unity, since the Galois group $\Gal(k'/k)$ of the extension $k'$ which is obtained by adjoining these roots is always solvable.

In practical applications of the solvability condition, a very important fact is that the Galois group of an equation can be computed without solving the equation itself. The idea of this computation can be stated as follows. Each automorphism of the splitting field of a polynomial $f$ induces a permutation of its roots, and the automorphism is completely determined by this permutation. For this reason the Galois group of the equation may be treated, in principle, as a subgroup of the group of permutations of its roots (namely, the subgroup consisting of permutations that preserve all algebraic relations between the roots). The relations between the roots of a polynomial yield certain relations between its coefficients (by virtue of Viète's formulas); by analyzing these relations it is possible to determine the relations between the roots of the polynomial and thus to compute the Galois group of the equation. In the general case the Galois group of an algebraic equation consists of all permutations of the roots, i.e. is the symmetric group of degree $n$. Since a symmetric group is unsolvable for $n\ge 5$, there are, generally speaking, no solutions in radicals of quintic equations and equations of a higher degree (Abel's theorem).

The ideas of Galois theory permit, in particular, to give a complete description of the class of construction problems that are solvable by ruler and compass. It is possible to show by methods of analytic geometry that any such construction problem can be reduced to some algebraic equation over the field of rational numbers, and the problem is solvable by using a ruler and compass if and only if the corresponding equation is solvable in quadratic radicals. This is the case if and only if the Galois group of the equation has a normal series whose factors are groups of order two, and this occurs if and only if its order is a power of two. Thus, a construction problem solvable by ruler and compass is reduced to solving an equation whose splitting field has degree $2^s$ over the field of rational numbers for some $s$; if the degree of the equation is not of the form $2^s$, such a construction is impossible. This is the case for the problem of doubling the cube (which leads to the cubic equation $x^3-2 = 0$) and for the problem of trisecting a given angle (which also leads to a cubic equation). If $p$ is a prime number, the problem of constructing a regular $p$-gon leads to the equation $x_{p-1}+\cdots + x + 1 = 0$, whose splitting field is generated by any root and is therefore of degree $p-1$, the degree of the equation. In this case a ruler-and-compass construction is possible if and only if $p=2^s+1$ (e.g. if $p=5$ and $p=17$ it is possible, but is not possible for $p=7$ or $p=13$).

The ideas of Galois had a decisive influence on the development of algebra during almost a whole century. Galois theory was extended and generalized in many directions. W. Krull developed a Galois theory for infinite extensions (cf. Galois topological group); it was proved (the Kronecker–Weber theorem) that the roots of an equation with rational coefficients and with an Abelian Galois group are rational linear combinations of roots of unity; a classification of the Abelian extensions of a given algebraic number field (class field theory) was given; a proof was given for the existence of an algebraic number field with given solvable Galois group over the field of rational numbers (cf. Galois theory, inverse problem of). Nevertheless, classical Galois theory still contains many unsolved problems. For instance, it is not known whether any finite group occurs as the Galois group of an equation over the field of rational numbers.

References

[1] E. Galois, "Écrits et mémoires d'E. Galois" , Gauthier-Villars (1962)
[2] N.G. [N.G. Chebotarev] Tschebotaröw, "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) (Translated from Russian)
[3] N.G. Chebotarev, "Galois theory" , Moscow-Leningrad (1936) (In Russian)
[4] M.M. Postnikov, "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)
[5] M.M. Postnikov, "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)
[6] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[7] S. Lang, "Algebra" , Addison-Wesley (1974)
[8] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)
[9] H. Koch, "Galoissche Theorie der $p$-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)
[10] E. Artin, "Galois theory" , Notre Dame Univ. , Indiana (1948)
How to Cite This Entry:
Galois theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_theory&oldid=19664
This article was adapted from an original article by M.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article